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Like the statistical mean and median, the mode is a way of expressing, in a (usually) single number, important information about a random variable or a population. The numerical value of the mode is the same as that of the mean and median in a normal distribution, and it may be very different in highly skewed distributions.
In fact, for a normal distribution, mean = median = mode. The median of a uniform distribution in the interval [a, b] is (a + b) / 2, which is also the mean. The median of a Cauchy distribution with location parameter x 0 and scale parameter y is x 0, the location parameter.
The most common measures of central tendency are the arithmetic mean, the median, and the mode. A middle tendency can be calculated for either a finite set of values or for a theoretical distribution, such as the normal distribution .
For example, the mean or average of the numbers 2, 3, 4, 7, and 9 (summing to 25) is 5. Depending on the context, the most representative statistic to be taken as the average might be another measure of central tendency, such as the mid-range, median, mode or geometric mean.
The mean of a set of observations is the arithmetic average of the values; however, for skewed distributions, the mean is not necessarily the same as the middle value (median), or the most likely value (mode). For example, mean income is typically skewed upwards by a small number of people with very large incomes, so that the majority have an ...
Average wage in the United States was $69,392 in 2020. [1] Median income per person in the U.S. was $42,800 in 2019. [2] The average is higher than the median because there are a small number of individuals with very high earnings, and a large number of individuals with relatively low earnings. (See Income inequality in the United States.)
Comparison of mean, median and mode of two log-normal distributions with different skewness. The mode is the point of global maximum of the probability density function. In particular, by solving the equation ( ln f ) ′ = 0 {\displaystyle (\ln f)'=0} , we get that:
While for a beta distribution with equal shape parameters α = β, it follows that skewness = 0 and mode = mean = median = 1/2, the geometric mean is less than 1/2: 0 < G X < 1/2. The reason for this is that the logarithmic transformation strongly weights the values of X close to zero, as ln( X ) strongly tends towards negative infinity as X ...